Separating fragments of bounded arithmetic

149 Pages·2009·0.72 MB·English

Arnold Beckmann
Separating fragments
of
bounded arithmetic
1996

Contents
1 Introduction 1
1.1 Bounded arithmetic . . . . . . . . . . . . . . . . . . . . . 1
1.2 Towards Dynamic Ordinals . . . . . . . . . . . . . . . . . 3
1.3 Extended summary . . . . . . . . . . . . . . . . . . . . . 5
2 Basic Definitions 13
3 Pure Number Theory 17
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 The well-ordering proof in IΣ0 . . . . . . . . . . . . . . . 19
n
4 Semi-formal Systems 25
4.1 The infinitary language . . . . . . . . . . . . . . . . . . . 25
4.2 The infinitary system . . . . . . . . . . . . . . . . . . . . 27
4.3 The Embedding of IΣ0 . . . . . . . . . . . . . . . . . . . 29
n
5 Upper Bounds for O(IΣ0) 33
n
5.1 L -cut-elimination . . . . . . . . . . . . . . . . . . . . . 33
ω
5.2 The Boundedness Theorem . . . . . . . . . . . . . . . . . 37
5.3 Applications: O(IΣ0 ) = ω (0) and O(IΣ0) = ω2 . . . 41
n+1 n+3 0
6 Notations for Exponentiation 45
6.1 Exponential codes for natural numbers . . . . . . . . . . 45
6.2 Limited course-of-values recursion . . . . . . . . . . . . . 48
6.3 E, ≺, +ˆ, T are polytime . . . . . . . . . . . . . . . . . . 50
E
7 Bounded Predicative Arithmetic (BPA) 57
7.1 The language . . . . . . . . . . . . . . . . . . . . . . . . 58
7.2 The theories . . . . . . . . . . . . . . . . . . . . . . . . . 60
i
ii CONTENTS
8 Bounded Arithmetic (BA) and BPA 67
8.1 Fragments of BA . . . . . . . . . . . . . . . . . . . . . . 67
8.2 Comparing theories of BA . . . . . . . . . . . . . . . . . 69
8.3 Comparing theories of BPA . . . . . . . . . . . . . . . . 72
8.4 Comparing BA with BPA . . . . . . . . . . . . . . . . . 73
9 Well-ordering Proofs in BPA 75
9.1 Formalization of wellfoundedness . . . . . . . . . . . . . 75
9.2 What pBASIC can prove . . . . . . . . . . . . . . . . . . 78
9.3 Proving foundation by induction . . . . . . . . . . . . . . 84
10 A Semi-formal System for BPA 87
10.1 Lp and the semi-formal system bsfp . . . . . . . . . . . 87
ω
10.2 The embedding into bsfp . . . . . . . . . . . . . . . . . 92
10.3 Extended cut-elimination I . . . . . . . . . . . . . . . . . 97
10.4 Extended cut-elimination II . . . . . . . . . . . . . . . . 99
11 Predicative Boundedness Theorems (PBT) 101
11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 101
11.2 Indiscernibles . . . . . . . . . . . . . . . . . . . . . . . . 102
11.3 Negative points and monotonicity . . . . . . . . . . . . . 104
11.4 Proving PBT . . . . . . . . . . . . . . . . . . . . . . . . 106
12 Dynamic Ordinal Analysis (DOA) 111
12.1 Dynamic Ordinals and separation . . . . . . . . . . . . . 112
12.2 Computing Dynamic Ordinals . . . . . . . . . . . . . . . 113
12.3 DOA in theories of BA . . . . . . . . . . . . . . . . . . . 117
A Standard interpretations 121
B pBASIC 123
C Proving indiscernibility 127
Acknowledgements
This work has been partially supported by a grant of the Graduierten-
stipendium of the Westf¨alische Wilhelms-Universit¨at Mu¨nster.
I am deeply indebted to Prof. Wolfram Pohlers, who roused
my interest in mathematical logic and tought me many of its domains.
He supervised this thesis and supported my plans for gaining the PhD.
I would like to thank Prof. Justus Diller of the Institut fu¨r Mathe-
matische Logik und Grundlagenforschung. He also had great influence
on me, in particular in several courses he lectured. In addition to this
I would like to thank the former and present members of staff and stu-
dentsoftheInstitut fu¨r Mathematische Logik und Grundlagenforschung
for the pleasant atmosphere in this institute. This greatly facilitated
my work on this dissertation. Above all I would like to thank HDoz.
Andreas Weiermann for his great understanding for my work and
his intensive cooperation.
Many thanks are due to Prof. Samuel R. Buss (San Diego), who,
in Bern in 1994, kindly paid attention to the early phase of my thesis
and put the question to me, if the results were also transferable to non-
relativized bounded predicative arithmetic. To Prof. Jan Kraj´ıcˇek
(Prague) and Prof. Pavel Pudla´k (Prague), who, in 1996, made me
feel very comfortable at Prague and listened to my lecture on the fi-
nal version of my thesis. To the following people from Mu¨nster for
comments on the manuscript: Benjamin Blankertz, Wolfgang
Burr, Finja Ku¨tz, Martina Pfeifer, Elke Sapho¨rster and
Kai F. Wehmeier.
iii
iv
Chapter 1
Introduction
The present work represents the author’s PhD-thesis at the Mathe-
matisch-Naturwissenschaftliche Fakult¨at of the Westf¨alische Wilhelms-
Universit¨at Mu¨nster, which has been developed under supervision of
Prof. W. Pohlers.
The aim of this work is to investigate proof-theoretically formal
theories of bounded arithmetic. For this purpose the subsystems IΣ0 of
n
firstorderarithmeticandsubsystemsofboundedpredicativearithmetic
will be investigated, too.
1.1 Bounded arithmetic
”Bounded arithmetic theories are subtheories of first order arithmetic.
They attempt to formalize reasoning about finite structures”1. In [6]
S. Buss introduced the theories Sn, Tn, U1, V1 of bounded arithmetic
2 2 2 2
which correspond to the computational classes in the polynomial time
hierarchy PH, PSPACE and EXPTIME. The classes P of languages
computable in polynomial time on deterministicTuring-machines and
NP of languages computable in polynomial time on non-deterministic
Turing-machines are levels of PH.
Itisacommonassumptionthattheseparationproblemsofbounded
arithmetic theories are essentially the same as the separation problems
ofcomputationalclasses(includingPvs.NP),althoughtheonlyknown
result in this relation is the following result in [16]:
Tn = Sn+1 =⇒ Σp = Πp .
2 2 n+2 n+2
1See [9] p. 2.
1
2 CHAPTER 1. INTRODUCTION
Thus, the collapse of S implies the collapse of PH.2 Therefore, the sep-
2
aration problems of bounded arithmetic theories are among the major
unsolved problems of the present time.
In the case of relativized computational classes things are quite
different. It has been shown in [1] that there are oracles A and B such
that PA = NPA and PB 6= NPB. In [25] and also in [13] it has been
shown that there is an oracle A such that PHA (i.e., the polynomial
time hierarchy with an oracle A) does not collapse.
Corresponding results for bounded arithmetic theories are proved by
usingtheseresults. ThesetΣb (X)ofbounded formulasofthelanguage
∞
of bounded arithmetic with set parameters X ,X ,... is stratified into
0 1
levels Σb(X) ⊂ Σb(X) ⊂ ... similar as the arithmetical formulas are
0 1
stratified into levels Σ0(X) ⊂ Σ0(X) ⊂ .... More precisely, Σb(X) is
0 1 0
the set of bounded formulas where all quantifiers are sharply bounded
quantifiers (i.e., they are bounded by a term of the form |t|, where
|n| = ⌈log (n+1)⌉). In addition to this Σb (X) is the set of bounded
2 i+1
formulas withi alternations of boundedquantifiers, which start with an
existential one and do not count the sharply bounded ones. The prenex
(or strict) versions of Σb(X) (where the closure under sharply bounded
i
quantifiers is omitted) are denoted by sΣb(X). The sets of bounded
i
formulas without set variables will be denoted omitting ”(X)”.
Let |y| ··= y and |y| ··= |(|y| )|. The theories Σb(X)-LmInd
0 m+1 m n
are axiomatized by a finite set of defining axioms for the non-logical
symbols and by the induction schema which consists of all formulas of
the form
F(0) ∧ ∀x<|t| (F(x) → F(x+1)) → F(|t| )
m m
with F ∈ Σb(X) and t being a term. As exponentiation λn.2n is not
n
a function which can be proved to be total in bounded arithmetic, this
induction schema seems to become weaker if m increases. The theories
with small numbers m have special names:
sRn(X) ··= sΣb(X)-L2Ind
2 n
Rn(X) ··= Σb(X)-L2Ind
2 n
Sn(X) ··= Σb(X)-L1Ind
2 n
Tn(X) ··= Σb(X)-L0Ind.
2 n
2Cf. [24].